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Proof of the Vector Triple Product - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Vector Triple Product is considered one the most difficult concept.

  • 35 Questions around this concept.

Solve by difficulty

The vectors \vec{a}\; and \; \vec{b} are not perpendicular and \vec{c}\; and \; \vec{d} are two vectors satisfying: \vec{b}\; \times \; \vec{c}=\vec{b}\; \times \; \vec{d}and \vec{a}\cdot \vec{d}=0. Then the vector \vec{d} is equal to

Let a,b and c be vectors then a vector which is such that, it is not perpendicular to c and a×b together is

\begin{array}{l}{\text { Let } \vec{a}=2 \hat{i}+\hat{j}-2 \hat{k} \text { and } \vec{b}=\hat{i}+\hat{j}} \\ {\text { Let } \vec{c} \text { be a vector such that }|\vec{c}-\vec{a}|=3,|(\vec{a} \times \vec{b}) \times \vec{c}|=3} \\ {\text { and the angle between } \vec{c} \text { and } \vec{a} \times \vec{b} \text { is } 30^{\circ} . \text { Then } \vec{a} . \vec{c} \text { is equal to }}\end{array}

a×(b+c)=

If a~=ijk,b~=ij+k and c~=i+2jk, then a×(b×c) equals

Let a=i^+2j^+3k^,b=3i^+j^k^ and c be three vectors such that c is coplanar with a and b. If the vector c is perpendicular to b and ac=5, then |c| is equal to

Concepts Covered - 1

Vector Triple Product

For three vectors a,b and c vector triple product is defined as a×(b×c). a×(b×c)=(ac)b(ab)c

p=a×(b×c) is a vector perpendicular to a and b×c, but b×c is a vector perpendicular to the plane of b and c. Hence, vector p must lie in the plane of b and c.
Let p=a×(b×c)=lb+mc[l,m are scalars ]
Taking the dot product of eq (i) with a, we get
pa=l(ab)+m(ac)

[a×(b×c) is aa×(b×c)a=0]

Therefore,
pa=0l(ab)=m(ac)1ac=mab=λl=λ(ac) and m=λ(ab)

Substituting the value of l and m in Eq. (i), we get
a×(b×c)=λ[(ac)b(ab)c]

Here, the value of λ can be determined by taking specific values of a,b and c.

The simplest way to determine λ is by taking specific vectors a=i^,b=i^,c=j^.
We have,
a×(b×c)=λ[(ac)b(ab)c]i^×(i^×j^)=λ[(i^j^)i^(i^i^)j^]i^×k^=λ[(0)i^(1)j^]j^=λj^λ=1

Hence,
a×(b×c)=(ac)b(ab)c

NOTE:

1.
a×(b×c)=(ac)b(ab)c(a×b)×c=(ca)b(cb)a
2. In general a×(b×c)(a×b)×c If a×(b×c)=(a×b)×c then the vectors a and c are collinear.

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Vector Triple Product

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